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Properties

Label 20.0.92250721279...8641.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{8}\cdot 293^{8}$
Root discriminant $31.50$
Ramified primes $19, 293$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_5$ (as 20T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -2048, 2816, -2688, 2240, -1600, 1040, -616, 316, -162, 87, -81, 79, -77, 65, -50, 35, -21, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 11*x^18 - 21*x^17 + 35*x^16 - 50*x^15 + 65*x^14 - 77*x^13 + 79*x^12 - 81*x^11 + 87*x^10 - 162*x^9 + 316*x^8 - 616*x^7 + 1040*x^6 - 1600*x^5 + 2240*x^4 - 2688*x^3 + 2816*x^2 - 2048*x + 1024)
 
gp: K = bnfinit(x^20 - 4*x^19 + 11*x^18 - 21*x^17 + 35*x^16 - 50*x^15 + 65*x^14 - 77*x^13 + 79*x^12 - 81*x^11 + 87*x^10 - 162*x^9 + 316*x^8 - 616*x^7 + 1040*x^6 - 1600*x^5 + 2240*x^4 - 2688*x^3 + 2816*x^2 - 2048*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 11 x^{18} - 21 x^{17} + 35 x^{16} - 50 x^{15} + 65 x^{14} - 77 x^{13} + 79 x^{12} - 81 x^{11} + 87 x^{10} - 162 x^{9} + 316 x^{8} - 616 x^{7} + 1040 x^{6} - 1600 x^{5} + 2240 x^{4} - 2688 x^{3} + 2816 x^{2} - 2048 x + 1024 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(922507212791997403369454768641=19^{8}\cdot 293^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $19, 293$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{5}{16} a^{10} - \frac{1}{8} a^{9} - \frac{3}{16} a^{8} + \frac{7}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{5}{32} a^{11} + \frac{7}{16} a^{10} + \frac{13}{32} a^{9} - \frac{9}{32} a^{8} + \frac{15}{32} a^{7} + \frac{15}{32} a^{6} - \frac{9}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{5}{64} a^{12} + \frac{7}{32} a^{11} + \frac{13}{64} a^{10} + \frac{23}{64} a^{9} - \frac{17}{64} a^{8} + \frac{15}{64} a^{7} + \frac{23}{64} a^{6} - \frac{1}{32} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{5}{128} a^{13} + \frac{7}{64} a^{12} + \frac{13}{128} a^{11} + \frac{23}{128} a^{10} + \frac{47}{128} a^{9} - \frac{49}{128} a^{8} + \frac{23}{128} a^{7} - \frac{1}{64} a^{6} + \frac{3}{16} a^{5} - \frac{7}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{617216} a^{18} - \frac{293}{77152} a^{17} + \frac{2355}{617216} a^{16} - \frac{8953}{617216} a^{15} - \frac{15889}{617216} a^{14} - \frac{1807}{308608} a^{13} + \frac{64569}{617216} a^{12} + \frac{104679}{617216} a^{11} + \frac{67739}{617216} a^{10} + \frac{7075}{617216} a^{9} - \frac{252693}{617216} a^{8} - \frac{19687}{308608} a^{7} - \frac{3467}{154304} a^{6} - \frac{6629}{19288} a^{5} - \frac{5257}{38576} a^{4} + \frac{8615}{19288} a^{3} + \frac{2355}{4822} a^{2} - \frac{2277}{4822} a + \frac{2}{2411}$, $\frac{1}{267871744} a^{19} - \frac{13}{19133696} a^{18} + \frac{865687}{267871744} a^{17} - \frac{1996139}{267871744} a^{16} - \frac{1077607}{267871744} a^{15} + \frac{416177}{66967936} a^{14} + \frac{539383}{38267392} a^{13} + \frac{4141391}{38267392} a^{12} + \frac{1351731}{8641024} a^{11} - \frac{128867763}{267871744} a^{10} + \frac{89794101}{267871744} a^{9} - \frac{783187}{2160256} a^{8} + \frac{2552445}{9566848} a^{7} - \frac{1440217}{4783424} a^{6} + \frac{3271801}{8370992} a^{5} + \frac{2014477}{8370992} a^{4} + \frac{1626435}{4185496} a^{3} + \frac{209660}{523187} a^{2} + \frac{55481}{149482} a + \frac{151644}{523187}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3891860.51282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

10.10.960472390437121.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
293Data not computed